Folks in Alaska know that they are living under special circumstance. It isn't a surprise to them that it is extra cold, for extra long.
Summer is coming, along with warmer weather.
First, as mentioned, check and make sure that his zip code offers overnight delivery. Anchorage should be fine, but double check. Many parts of Alaska are 2 day or longer delivery, regardless of the overnight booking.
Second, wait until the weather is acceptable for live reptile shipping period. If that is next week, great. If it isn't until mid-June, then so be it. Default to the health and welfare of the animals.
Again, living in Alaska, the customer should be familiar with the limitations.
Money wise, I would offer an immediate refund, good for a two day consideration. If the customer doesn't want to wait for a window of appropriate weather, go ahead and refund now. Give them two days to consider that option.
If they pass and say "let's wait for better weather" then any payment would be treated like a non-refundable deposit, and you would keep any monies paid if they change their mind down the road.
You don't want to take payment, mark the animal sold, then get told in mid-June, "Oh, I changed my mind, I want a refund." That is poor form on behalf of the buyer, so consider covering yourself for that circumstance.
At ShipYourReptiles we book Alaska shipments regularly, but by phone instead of through the site. This allows Andy@SYR to check for delivery guarantee times and also help evaluate weather. Feel free to contact
Andy@ShipYourReptiles.com for help in arranging this shipment.